Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #142568 |
| commutator, Frattini, cosocle | #2309 | C23 |
| prime radical | C22×H(5,𝔽2) | D4 |
| LCS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G | |
| LECS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1288 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 1288 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,12)(3,13,16,40)(4,9,21,34)(5,10,23,35)(6,11,26,37)(8,27,30,17)(14,41,43,38)(15,39,22,55)(18,48,50,63)(19,51,52,32)(20,54,25,33)(24,36,45,62)(28,57,59,44)(29,46,60,47)(31,56,61,49)(42,64,53,58);(1,3,6,17)(2,8,11,13)(4,18,20,49,21,50,25,56)(5,22,24,47)(7,16,26,27)(9,31,33,48,34,61,54,63)(10,29,36,55)(12,30,37,40)(14,42,44,19,43,53,57,52)(15,45,46,23)(28,58,41,51,59,64,38,32)(35,60,62,39);(1,4)(2,9)(3,14)(5,19)(6,25)(7,21)(8,28)(10,32)(11,33)(12,34)(13,38)(15,18)(16,43)(17,44)(20,26)(22,50)(23,52)(24,42)(27,57)(29,31)(30,59)(35,51)(36,58)(37,54)(39,48)(40,41)(45,53)(46,49)(47,56)(55,63)(60,61)(62,64)Polycyclic presentation:
2^10|a^2=j,b^2=f,f^2=h,b^a=bf,c^a=ce,c^b=cd,d^a=dh,d^b=dg,e^b=egh,e^d=ej,f^a=fh,f^c=fg,f^d=fij,f^e=fij,g^a=gi,g^b=gij,g^f=gj,h^c=hj,i^b=ij46 conjugacy classes — view the full character table.
| maximal subgroup of | #72285086, #72285854, #72285936 |
| maximal normal subgroup of | #72285086, #72285854, #72285936 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);